1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 25 numbers other than -1 can be put into one-to-one correspondence with the real numbers. 8. If u and v are unim ...

26 Chapter^1 and then prove that lzl - Re z :5^1 / 2 iff z = ac with le - al :5 1. [S. I. Drobnies, Amer. Math. Monthly, 71 (197 ...

Complex Numbers 27 ( c) Show that the equation represents the point -b if a = b, c is real, and lbl^2 - c = 0, and has no .solut ...

28 Chapter 1 z = (a, b) = a + bi is made to correspond to the point P with coordinates x = a, y = b, and conversely. This proced ...

Complex Numbers 29 correspondence between the complex numbers and the vectors with the origin 0, provided that we associate the ...

30 Chapter^1 Since the trigonometric functions cos () and sin() can be defined analytically by means of the power series expansi ...

Complex Numbers 31 If z = J3-i, we haver= .J3+T = 2, cosB = J3/2, sinB = -1/2, so that 00 = -7r /6. Hence J3 -i = 2 [cos ( ~7r ...

32 Chapter 1 (1.8-4) The two preceding equations imply that (Ji = 82 + 2k7r, since there is just one value of 8 on any semiopen ...

Complex Numbers 33 Theorem 1. 7 The product of two nonzero complex numbers is a complex number whose modulus is the product of t ...

34 so that and Note If 81 = Arg z1, 82 = Arg z2, then Z1 Arg - = Argz1-Argz2+27rn(z1,z2) Z2 where Exercises 1.3 if - 7l" < Ar ...

Complex Numbers 35 i + V3 (1-i)^2 (c) (l+i)2 (d) J3-i Given z = (1 +cos()+ i sin B)/(1 +cos()' + i sin()'), find: (a) lzl (b) ...

36 Chapter^1 the segment OC by OC, that of OB by OB, and so on, we have OC = OB+ BC= OB+ OA =a+ c CR= CD+DR= BP+AQ = b+d which p ...

Complex Numbers 37 y x Fig. 1.6 If we let z 1 =(a, b) and z 2 = (c, d), we have lz1 - z2I = v(a - c)^2 + (b - d)^2 =length of QP ...

38 Chapter^1 we proceed as follows (see Fig. 1.7). Suppose that OA is the vector corre- sponding to the number (1, 0), and consi ...

Complex Numbers D I I I I 0 A Fig. 1.9 Q I I \ I \ \ p 39 A geometric construction of the inverse of a given point with respect ...

40 Chapter^1 x Fig. 1.11 EXERCISES 1.4 Do alf operations geometrically. (a) (-3,4)+(-2,-5) (b) (1+2i)+(3-4i) (c) (4+i)+(1+3i)+ ...

Complex Numbers 41 Show that if the four points z 1 , z 2 , z 3 , z 4 are the vertices of a convex quadrilateral, then where e ...

42 Chapter^1 ( e) Again assuming that z 2 and z 3 are fixed, find the locus of z1 when ). = eit ( t real). [Gae. Mat., 20 (1968) ...

Complex Numbers by repeated application of Theorem 1. 7, we find that Z1Z2 ... Zn= (r1r2 ... rn)ei(81+82+··+8n) 43 In particular ...

44 y 0 Fig. 1.12 P4(z4) I I I I I P3(z3) I I I I P2(z2) I I I I P1(Z) Chapter 1 x This value of the argument of i^3 (i.e., the p ...